Let {X} be a set. Let {\mathcal{G}} be a non-empty collection of subsets of {X} such that {\mathcal{G}} is closed under finite intersections. Assume that there exists a sequence {X_h \in \mathcal{G}} such that {X = \cup_h X_h}. Let {\mathcal{M}} be the smallest collection of susbsets of {X} containing {\mathcal{G}} such that the following are true:

If {E_h \in \mathcal{M}} {\forall h \in \mathbb{N}} and {E_h} {\uparrow} {E} then {E \in \mathcal{M}}
If {E}, {F}, {E \cup F \in \mathcal {M}} then {E \cap F \in \mathcal {M}}
If {E \in \mathcal {M}} then {E^c \in \mathcal{M}}

Does {X \in \mathcal{M}} ?